Inductance of solenoids

Transcription

1 Inductance of solenoids LEP -01 Related topics Law of inductance, Lenz s law, self-inductance, solenoids, transformer, oscillatory circuit, resonance, damped oscillation, logarithmic decrement, Q factor. Principle A square wave voltage of low frequency is applied to oscillatory circuits comprising coils and capacitors to produce free, damped oscillations. The values of inductance are calculated from the natural frequencies measured, the capacitance being known. Equipment Induction coil, 300 turns, d = 40 mm Induction coil, 300 turns, d = 3 mm Induction coil, 300 turns, d = 5 mm Induction coil, 00 turns, d = 40 mm Induction coil, 100 turns, d = 40 mm Induction coil, 150 turns, d = 5 mm Induction coil, 75 turns, d = 5 mm Coil, 100 turns Oscilloscope, 30 MHz, channels Function generator Capacitor /case 1/ 470 nf Adapter, BNC-plug/socket 4 mm Connection box Connecting cord, l = 50 mm, red Connecting cord, l = 500 mm, red Connecting cord, l = 50 mm, blue Connecting cord, l = 500 mm, blue Tasks To connect coils of different dimensions (length, radius, number of turns) with a known capacitance C to form an oscillatory circuit. From the measurements of the natural frequencies, to calculate the inductances of the coils and determine the relationships between 1. inductance and number of turns. inductance and length 3. inductance and radius. Set-up and procedure Set up the experiment as shown in Fig A square wave voltage of low frequency (f 500 Hz) is applied to the excitation coil L. The sudden change in the magnetic field induces a voltage in coil L1 and creates a free damped oscillation in the L1C oscillatory circuit, the frequency f o of which is measured with the oscilloscope. Coils of different lengths l, diameters r and number of turns N are available (Tab. 1). The diameters and lengths are measured with the vernier caliper and the measuring tape, and the numbers of turns are given. Fig. 1: Experimental set-up. PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

2 LEP -01 Inductance of solenoids Fig. : Set-up for inductance measurement. The following coils provide the relationships between inductance and radius, length and number of turns that we are investigating: 1.) 3, 6, 7 L = f(n).) 1, 4, 5 L/N = f(l) 3.) 1,, 3 L = f(r) As a difference in length also means a difference in the number of turns, the relationship between inductance and number of turns found in Task 1 must also be used to solve Task. Tab. 1: Table of coil data Coil No. N r l mm mm Cat. No Fig. 3 shows an measurement example and the oscilloscope settings: - Input: CH1 - Volts/div 10 mv - Time/div <0.1 ms - Trigger source CH1 - Trigger mode Norm The oscilloscope shows the rectangular signal and the damped oscillation behind each peak. Determine the frequency f 0 of this damped oszillation. f 0 1 T where T is the oscillation period. Fig. 3: Oscilloscope settings PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

3 Inductance of solenoids LEP -01 Fig. 4: Inductances of the coils as a function of the number of turns, at constant length and constant radius. Double logarithmic plotting. Fig. 6: Inductance of the coils as a function of the radius, at constant length and number of turns. Double logarithmic plotting. Notes The distance between L1 and L should be as large as possible so that the effect of the excitation coil on the resonant frequency can be disregarded. There should be no iron components in the immediate vicinity of the coils. The tolerance of the oscilloscope time-base is given as 4%. If a higher degree of accuracy is required, the time-base can be calibrated for all measuring ranges with the function generator and a frequency counter prior to these experiments. The magnetic flux through the coil is given by = m o m H A () where m o is the magnetic field constant and m the absolute permeability of the surrounding medium. When this flux changes, it induces a voltage between the ends of the coil, U ind. N Theory and evaluation If a current of strength I flows through a cylindrical coil (solenoid) of length l, cross sectional area A = p r, and number of turns N, a magnetic field is set up in the coil. When l >> r the magnetic field is uniform and the field strength H is easy to calculate: H I N l (1) where N m 0 m A N l I L I L m 0 m p N r l (3) (4) is the coefficient of self-induction (inductance) of the coil. Inductivity Equation (4) for the inductance applies only to very long coils l >> r, with a uniform magnetic field in accordance with (1). In practice, the inductance of coils with l > r can be calculated with greater accuracy by an approximation formula L N r a r l b 3>4 for 0 6 r (5) l 6 1 Fig. 5: Inductance per turn as a function of the length of coil, at constant radius. Double logarithmic plotting. In the experiment, the inductance of various coils is calculated from the natural frequency of an oscillating circuit. v 0 1 LC tot. C tot. is the sum of the capacitance the known capacitor and the input capacitance C i of the oscilloscope. (6) PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

4 LEP -01 Inductance of solenoids The internal resistance R i of the oscilloscope exercises a damping effect on the oscillatory circuit and causes a negligible shift (approx. 1%) in the resonance frequency. The inductance is therefore represented by 1 L 4p f 0 C tot. where C tot. = C + C i and The table shows the theoretical inductance values of the used coils calculated according to eq. 5. Table f 0 v 0 p Coil No. N r/m l/m L theo /µh The table 3 shows the measured values of the oscillation periods and the corresponding inductance values of the used coils calculated according to eq. 7. These L exp values are plotted in Figs. 4, 5 and 6. (7) L = A N B to the regression line from the measured values in Fig. 4 gives B = 1.95±0.04 ; B theo = (see Eq. 5) Now that we know that L ~ N, we can demonstrate the relationship between inductance and the length of the coil. L A lc N to the regression line from the measured values in Fig. 5 gives C = 0.8 ± ; C theo = L A rd N to the regression line from the measured values in Fig. 6 gives D = 1.86 ± ; D theo = 1.75 The Equation (5) is thus verified within the limits of error. Table 3 Coil No. T exp./µs L exp /µh PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

5 Inductance of solenoids with Cobra3 LEP -11 Related topics Law of inductance, Lenz s law, self-inductance, solenoids, transformer, oscillatory circuit, resonance, damped oscillation, logarithmic decrement, Q factor. Principle A square wave voltage of low frequency is applied to oscillatory circuits comprising coils and capacitors to produce free, damped oscillations. The values of inductance are calculated from the natural frequencies measured, the capacitance being known. Equipment Cobra3 Basic Unit Power supply, 1 V RS 3 data cable Cobra3 Universal writer software Cobra3 Function generator module Induction coil, 300 turns, dia. 40 mm Induction coil, 300 turns, dia. 3 mm Induction coil, 300 turns, dia. 5 mm Induction coil, 00 turns, dia. 40 mm Induction coil, 100 turns, dia. 40 mm Induction coil, 150 turns, dia. 5 mm Induction coil, 75 turns, dia. 5 mm Coil, 100 turns PEK capacitor /case 1/ 470 nf/50 V Connection box Connecting cord, 50 mm, red Connecting cord, 50 mm, blue Connecting cord, 500 mm, red Connecting cord, 500 mm, blue PC, Windows 95 or higher Tasks To connect coils of different dimensions (length, radius, number of turns) with a known capacitance C to form an oscillatory circuit. From the measurements of the natural frequencies, to calculate the inductances of the coils and determine the relationships between 1. inductance and number of turns. inductance and length 3. inductance and radius. Set-up and procedure Set up the experiment as shown in Fig A square wave voltage of low frequency (f 500 Hz) is applied to the excitation coil L. The sudden change in the magnetic field induces a voltage in coil L1 and creates a free damped oscillation in the L1C oscillatory circuit, the frequency f o of which is measured with the Cobra3 interface. Coils of different lengths l, diameters r and number of turns N are available (Tab. 1). The diameters and lengths are measured with the vernier caliper and the measuring tape, and the numbers of turns are given. Fig. 1: Experimental set-up. PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

6 LEP -11 Inductance of solenoids with Cobra3 Fig. : Set-up for inductance measurement. Fig. 3: Measuring parameters. Tab. 1: Table of coil data r l Coil No. N Cat. No. mm mm The following coils provide the relationships between inductance and radius, length and number of turns that we are investigating: 1.) 3, 6, 7 L = f(n).) 1, 4, 5 L/N = f(l) 3.) 1,, 3 L = f(r) As a difference in length also means a difference in the number of turns, the relationship between inductance and number of turns found in Task 1 must also be used to solve Task. Notes The distance between L1 and L should be as large as possible so that the effect of the excitation coil on the resonant frequency can be disregarded. There should be no iron components in the immediate vicinity of the coils. Connect the Cobra3 Basic Unit to the computer port COM1, COM or to USB port (for USB computer port use USB to RS3 Converter ). Start the measure program and select Cobra3 Universal Writer Gauge. Begin the measurement using the parameters given in Fig. 3. For the measurement of the oscillation period the Survey Function of the Measure Software is used (see Fig. 4). Fig. 4: Measurement of the oscillation period with the Survey Function PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

7 Inductance of solenoids with Cobra3 LEP -11 Fig. 4 shows the rectangular signal and the damped oscillation behind each peak. Determine the frequency f 0 of this damped oszillation, The inductance is therefore represented by 1 L 4p f 0 C tot. (7) where T is the oscillation period. Theory and evaluation If a current of strength I flows through a cylindrical coil (solenoid) of length l, cross sectional area A = p r, and number of turns N, a magnetic field is set up in the coil. When l >> r the magnetic field is uniform and the field strength H is easy to calculate: The magnetic flux through the coil is given by (1) = m o m H A () where m o is the magnetic field constant and m the absolute permeability of the surrounding medium. When this flux changes, it induces a voltage between the ends of the coil, where f 0 1 T H I N l U ind. N N m 0 m A N l I L I L m 0 m p N r is the coefficient of self-induction (inductance) of the coil. Inductivity Equation (4) for the inductance applies only to very long coils l >> r, with a uniform magnetic field in accordance with (1). In practice, the inductance of coils with l > r can be calculated with greater accuracy by an approximation formula L N r a r l b 3>4 l (3) (4) where C tot. = C + C i and The table shows the theoretical inductance values of the used coils calculated according to eq. 5. Table The table 3 shows the measured values of the oscillation periods and the corresponding inductance values of the used coils calculated according to eq. 7. These L exp values are plotted in Figs. 5, 6 and 7. Table 3 f 0 v 0 p Coil No. N r/m l/m L theo /µh Coil No. T exp. /µs L exp /µh Fig. 5: Inductances of the coils as a function of the number of turns, at constant length and constant radius. Double logarithmic plotting for 0 6 r (5) l 6 1 In the experiment, the inductance of various coils is calculated from the natural frequency of an oscillating circuit. v 0 1 LC tot. C tot. is the sum of the capacitance the known capacitor and the input capacitance C i of the Cobra3 input. The internal resistance R i of the Cobra3 input exercises a damping effect on the oscillatory circuit and causes a negligible shift (approx. 1%) in the resonance frequency. (6) PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen

8 LEP -11 Inductance of solenoids with Cobra3 Fig. 6: Inductance per turn as a function of the length of coil, at constant radius. Double logarithmic plotting Fig. 7: Inductance of the coils as a function of the radius, at constant length and number of turns. Double logarithmic plotting L = A N B to the regression line from the measured values in Fig. 5 gives B = 1.95±0.04 ; B theo = (see Eq. 5) Now that we know that L ~ N, we can demonstrate the relationship between inductance and the length of the coil. L A lc N to the regression line from the measured values in Fig. 6 gives C = -0.8±0.04 ; C theo = L A rd N to the regression line from the measured values in Fig. 7 gives D = 1.86 ± ; D theo = 1.75 The Equation (5) is thus verified within the limits of error PHYWE series of publications Laboratory Experiments Physics PHYWE SYSTEME GMBH & Co. KG D Göttingen